Statistics Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval

Where We’ve Been Populations are characterized by numerical measures called parameters Decisions about population parameters are based on sample statistics Inferences involve uncertainty reflected in the sampling distribution of the statistic McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

Where We’re Going Estimate a parameter based on a large sample Use the sampling distribution of the statistic to form a confidence interval for the parameter Select the proper sample size when estimating a parameter McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.1: Identifying the Target Parameter The unknown population parameter that we are interested in estimating is called the target parameter. Parameter Key Word or Phrase Type of Data µ Mean, average Quantitative p Proportion, percentage, fraction, rate Qualitative McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used to estimate the population parameter. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean Suppose a sample of 225 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours. What can we conclude about all college students’ television time? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean Assuming a normal distribution for television hours, we can be 95%* sure that *In the standard normal distribution, exactly 95% of the area under the curve is in the interval -1.96 … +1.96 McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean An interval estimator or confidence interval is a formula that tell us how to use sample data to calculate an interval that estimates a population parameter. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter. The confidence level is the confidence coefficient expressed as a percentage. (90%, 95% and 99% are very commonly used.) 95% sure McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean The area outside the confidence interval is called  95 % sure So we are left with (1 – 95)% = 5% =  uncertainty about µ McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean If  is unknown and n is large, the confidence interval becomes McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population Mean For the confidence interval to be valid … the sample must be random and … the sample size n must be large. If n is large, the sampling distribution of the sample mean is normal, and s is a good estimate of  McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.3: Small-Sample Confidence Interval for a Population Mean Large Sample Small Sample Sampling Distribution on  is normal Known  or large n Standard Normal (z) Distribution Sampling Distribution on  is unknown Unknown  and small n Student’s t Distribution (with n-1 degrees of freedom) McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.3: Small-Sample Confidence Interval for a Population Mean Large Sample Small Sample McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.3: Small-Sample Confidence Interval for a Population Mean For the confidence interval to be valid … the sample must be random and … the population must have a relative frequency distribution that is approximately normal* * If not, see Chapter 14 McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.3: Small-Sample Confidence Interval for a Population Mean Suppose a sample of 25 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours. What can we conclude about all college students’ television time? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.3: Small-Sample Confidence Interval for a Population Mean Assuming a normal distribution for television hours, we can be 95% sure that McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.4: Large-Sample Confidence Interval for a Population Proportion Sampling distribution of The mean of the sampling distribution is p, the population proportion. The standard deviation of the sampling distribution is where For large samples the sampling distribution is approximately normal. Large is defined as McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.4: Large-Sample Confidence Interval for a Population Proportion Sampling distribution of McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.4: Large-Sample Confidence Interval for a Population Proportion We can be 100(1-)% confident that where and McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.4: Large-Sample Confidence Interval for a Population Proportion A nationwide poll of nearly 1,500 people … conducted by the syndicated cable television show Dateline: USA found that more than 70 percent of those surveyed believe there is intelligent life in the universe, perhaps even in our own Milky Way Galaxy. What proportion of the entire population agree, at the 95% confidence level? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.4: Large-Sample Confidence Interval for a Population Proportion If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results where McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.4: Large-Sample Confidence Interval for a Population Proportion Suppose in a particular year the percentage of firms declaring bankruptcy that had shown profits the previous year is .002. If 100 firms are sampled and one had declared bankruptcy, what is the 95% CI on the proportion of profitable firms that will tank the next year? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size To be within a certain sampling error (SE) of µ with a level of confidence equal to 100(1-)%, we can solve for n: McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size The value of  will almost always be unknown, so we need an estimate: s from a previous sample approximate the range, R, and use R/4 Round the calculated value of n upwards to be sure you don’t have too small a sample. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size To estimate p, use the sample proportion from a prior study, or use p = .5. Round the value of n upward to ensure the sample size is large enough to produce the required level of confidence. For a confidence interval on the population proportion, p, we can solve for n: McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.5: Determining the Sample Size How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p? McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals